Linear Programming Models in Services

Learning Objectives Describe the features of constrained optimization models. Formulate LP models for computer solution. Solve two-variable models using graphics. Explain the nature of sensitivity analysis. Solve LP models with Excel Add-in Solver and interpret the results. Formulate a goal programming model.

Stereo Warehouse Let x = number of receivers to stock y = number of speakers to stock Maximize 50x + 20y gross profit Subject to 2x + 4y 400 floor space 100x + 50y 8000 budget x 60 sales limit x, y 0

Diet Problem Lakeview Hospital Let E = units of egg custard base in the shake C = units of ice cream in the shake S = units of butterscotch syrup in the shake Minimize Subject to cholesterol fat protein calories

Shift-Scheduling Problem Gotham City Police Patrol Let xi = number of officers reporting at period i for i =1, 2, 3, 4, 5, 6 Minimize x1 + x6 6 period 1 x1 + x2 4 period 2 x2 + x3 14 period 3 x3 + x4 8 period 4 x4 + x5 12 period 5 x5 + x6 16 period 6

Workforce-Planning Problem Last National Drive-in Bank Let Tt = number of trainees hired at the beginning of period t for t = 1,2,3,4,5,6 At = number of tellers available at the beginning of period t Minimize subject to A1 = 12 for t = 2,3,4,5,6 At , Tt 0 and integer for t = 1,2,3,4,5,6

Transportation Problem Lease-a-Lemon Car Rental Let xij = number of cars sent from city i to city j for i = 1,2,3 and j = 1,2,3,4 Minimize 439x11 + 396 x12 + . . . +479x33 + 0x34 subject to x11 + x12 + x13 + x14 = 26 x21 + x22 + x23 + x24 = 43 x31 + x32 + x33 + x34 = 31 x11 + x21 + x31 = 32 x12 + x22 + x32 = 28 x13 + x23 + x33 = 26 x14 + x24 + x34 = 14 xij 0 for all i , j

Graphical Solution Stereo Warehouse Z=3800 Z=3600 Z=3000 Z=2000 E Optimal solution ( x = 60, y = 40) D C A B

Model in Standard Form Let s1 = square feet of floor space not used s2 = dollars of budget not allocated s3 = number of receivers that could have been sold Maximize Z = 50x + 20y subject to 2x + 4y + s1 = 400 (constraint 1) 100x + 50y + s2 = 8000 (constraint 2) x + s3 = 60 ( constraint 3) x, y, s1, s2, s3 0

Stereo Warehouse Extreme-Point Solutions Extreme Nonbasic Basic Variable Objective-function point variables variables value value Z A x, y s1 400 0 s2 8000 s3 60 B s3, y s1 280 3000 s2 2000 x 60 C s3, s2 s1 120 3800 y 40 D s1, s2 s3 20 3600 y 80 x 40 E s1, x s3 60 2000 y 100 s2 3000

Sensitivity Analysis Objective-Function Coefficients z = 50x + 20y (constraint 3 ) D (constraint 1) (constraint 2) C A B

Sensitivity Analysis Right-Hand-Side Ranging (constraint 3 ) D H (constraint 2) C A B I

Goal Programming Stereo Warehouse Example Let x = number of receivers to stock y = number of speakers to stock = amount by which profit falls short of $99,999 = amount by which profit exceeds $99,999 = amount by which floor space used falls short of 400 square feet = amount by which floor space used exceeds 400 square feet = amount by which budget falls short of $8000 = amount by which budget exceeds $8000 = amount by which sales of receivers fall short of 60 = amount by which sales of receivers exceed 60 = priority level with rank k Minimize subject to profit goal floor-space goal budget goal sales-limit goal

Topics for Discussion How can the validity of LP models be evaluated? Interpret the meaning of the opportunity cost for a nonbasic decision variable that did not appear in the LP solution. Explain graphically what has happened when a degenerate solution occurs in an LP problem. Is LP a special case of goal programming? Explain. What are some limitations to the use of LP?